Simulation-based Bayesian optimal ALT designs for model discrimination

Accelerated life test (ALT) planning in Bayesian framework is studied in this paper with a focus of differentiating competing acceleration models, when there is uncertainty as to whether the relationship between log mean life and the stress variable is linear or exhibits some curvature. The proposed criterion is based on the Hellinger distance measure between predictive distributions. The optimal stress-factor setup and unit allocation are determined at three stress levels subject to test-lab equipment and testduration constraints. Optimal designs are validated by their recovery rates, where the true, datagenerating, model is selected under the DIC (Deviance Information Criterion) model selection rule, and by comparing their performance with other test plans. Results show that the proposed optimal design method has the advantage of substantially increasing a test plan's ability to distinguish among competing ALT models, thus providing better guidance as to which model is appropriate for the follow-on testing phase in the experiment. & 2014 Elsevier Ltd. All rights reserved. 1. Motivation for work Most work of the optimal Accelerated Life Testing (ALT) designs in the literature has focused on finding test plans that allowmore precise estimate of a reliability quantity, such as life percentile, at a lower stress level (it is usually the use stress level); see, for example, Nelson and Kielpinski [1] and Nelson and Meeker [2]. Nelson [3,4] summarized the ALT literature up to 2005 and a significant portion of this article is devoted to the optimal design of ALT planning. More recent discussions of optimal ALT plans and/or robust ALT plans can be found in, e.g., Xu [5], McGree and Eccleston [6], Monroe et al. [7], Yang and Pan [8], Konstantinou et al. [9], and Haghighi [10]. In the previous study, the associated confidence intervals of an estimate reflect the uncertainty arising from limited sample size and censoring at test, but do not account for model form inadequacy. However, model errors can be quickly amplified and potentially dominate other sources of errors in reliability prediction through the model-based extrapolation that characterizes ALTs. Implicit in the design criteria used in current ALTs is the assumption that the form of the acceleration model is correct. In many real-world problems this assumption could be unrealistic. A more realistic goal of an initial stage of ALT experimentation is to find an optimal design that helps in selecting a model among rival or competingmodel forms. The ALT designs that are good for model form discrimination could be quite different from those that are more appropriate for life percentile prediction under a specific model. Extrapolation in both stress and time is a typical characteristic of ALT inference. The most common accelerated failure time regression models (based, for example, on Lognormal or Weibull fit to the failure time distribution at a given stress level) are only adequate for modeling some simple chemical processes that lead to failure (Meeker and Escobar [11]). However, for modern electronic devices, more sophisticated models with basis in the physics of failure mechanisms are required. These complicated models are expected to have more parameters with possible interactions among stress factors. Therefore, investigating ALT designs with model selection capability is needed more than ever before. Meeker et al. [12] in their discussion of figures of merit when developing an ALT plan emphasizes the usefulness of a test plan's robustness to the departure from the assumed model. For example, when planning a single-factor experiment under a linear model, it is useful to evaluate the test plan properties under a quadratic model. Also, when planning a two-factor experiment under the assumption of a linear model with no interaction, it is useful to evaluate the test plan properties under a linear model with an interaction term. We strongly believe that it is worthwhile to consider these recommended practices ahead of time when the test plan is being devised in the first place by allowing a design criterion that is capable of model form discrimination.